On the graphs with distinguishing number equal list distinguishing number

Document Type : Research Paper

Authors

Department of Mathematical Sciences, Yazd University, Yazd, Iran

Abstract

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by the trivial automorphism. A list assignment to $G$ is an assignment $L = \{L(v)\}_{v\in V (G)}$ of lists of labels to the vertices of $G$. A distinguishing $L$-labeling of $G$ is a distinguishing labeling of $G$ where the label of each vertex $v$ comes from $L(v)$. The list distinguishing number of $G$, $D_l(G)$ is the minimum $k$ such that every list assignment to $G$ in which $|L(v)| = k$ for all $v \in V (G)$ yields a distinguishing $L$-labeling of $G$. In this paper, we determine the list-distinguishing number for two families of graphs. We also characterize graphs with the distinguishing number equal the list distinguishing number. Finally, we show that this characterization works for other list numbers of a graph.

Keywords


[1] M.O. Albertson and K.L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1996), #R18.
[2] S. Alikhani and S. Soltani, Distinguishing number and distinguishing index of certain graphs, Filomat, 31:14 (2017), 4393-–4404.
[3] F. Bonomo, G. Durán, J. Marenco, Exploring the complexity boundary between coloring and list-coloring, Ann. Oper. Res. 169 (2009) 3-16.
[4] M. Chan, The distinguishing number of the augmented cube and hypercube Powers, Discrete Math. 308 (2008), 2330-2336.
[5] L. S. Chandran, S. Padinhatteeri, K. Ravi Shankar, List Distinguishing Number of pth Power of Hypercube and Cartesian Powers of a Graph. In: Changat M., Das S. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2020. Lecture Notes in Computer Science, vol 12016. Springer, Cham.
[6] C.T. Cheng, On computing the distinguishing numbers of trees and forests, Electron. J. Combin. 13 (2006), #R11.
[7] K.L. Collins and A.N. Trenk, The distinguishing chromatic number, Electron. J. Combin. 13 (1) (2006), #R16.
[8] P. Erdős, A.L. Rubin and H. Taylor, Choosability in graphs, Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing, Congr. Numer. 26 (1979), 125-157.
[9] M. Ferrara, B. Flesch, E. Gethner, List-distinguishing colorings of graphs, Electron. J. Combin. 18 (2011), #P161.
[10] M. Ferrara, E. Gethner, S.G. Hartke, D. Stolee and P.S. Wenger, List distinguishing parameters of trees, Discrete Appl. Math. 161 (6) (2013), 864-869.
[11] R. Hammack, W. Imrich and S. Klav̌zar, Handbook of product graphs (second edition), Taylor & Francis group (2011).
[12] P. Immel and P.S. Wenger, The list distinguishing number equals the distinguishing number for interval graphs, Discuss. Math. Graph Theory. 37 (2017), 165-174.
[13] T.R. Jensen and B. Toft, Graph coloring problems, A Wiley-Interscience Publication, 1995.
[14] R. Kalinowski and M. Pilśniak, Distinguishing graphs by edge colourings, European J. Combin. 45 (2015) 124-131.
[15] D. Kim, Y.S. Kwon and J. Lee, The distinguishing numbers of Merged Johnson graphs, Bull. Korean Math. Soc. 52 (2015), 395-408.
[16] M. Lucci, G. Nasini, D. Seveŕın, A branch and price algorithm for list coloring problem, Elec. Notes Theoretical Comp. Sci. 346 (2019) 613-624.
[17] S.M. Smith and M.E. Watkins, Bounding the distinguishing number of infinite graphs and permutation groups, Electron. J. Combin. 21 (2014), #P3.40.
[18] W. Wang, X. Liu, List-coloring based channel allocation for open-spectrum wireless networks, IEEE 62nd Vehicular Technology Conference, Dallas, TX, USA (2005), 690-–694.